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Int Jr. of Mathematics Sciences & Applications Vol. 2, No. 2, May 2012 Copyright Mind Reader Publications ISSN No: 2230-9888 www.journalshub.com PHARMACOKINETIC AND OPTIMIZATION MODEL FOR PHOTODYNAMIC THERAPY B. K. Singh Deptt. of Mathematics IFTM University, Moradabad Paras Bhatnagar Deptt. of Mathematics Ideal Institute of Technology, Ghaziabad Chitra Gupta Deptt. of Mathematics IFTM University, Moradabad ABSTRACT Photodynamic therapy (PDT) is an emerging treatment that destroys undesirable tissues and is based on the idea that chemical compounds, called photo sensitizers, cause cellular damage when activated by light. Photo sensitizer exists in a variety of molecular compositions and structures, and different compositions react to different wavelength of light. When a photo sensitizer is activated by light, an electron is excited from a ground state to a more reactive state, converting it to a more energetic species. The photosensitizer we consider is photofrin, which is the only FDA approved photosensitizer for cancer treatment. While other photosensitizers are being clinically tested, accurate pharmacokinetic data is not available for these photosensitizers. Photofrin is intravenously injected at 2mg/kg over a period of five minutes, and it attains a high saturation level from 48 to 72 hours after injection. The 24 hour period between the 49th and 72nd hour is called the treatment window, and at some time within this period a physician focuses a 630nm light on the targeted area. After treatment, a patient is asked to avoid light exposure for six weeks. Keywords:photofrin, photosensitizer, first order linear differential equation, optimization model for photofrin. INTRODUCTION: - Our pharmacokinetic model estimates a region's concentration by first calculation the arterial concentration and second using this concentration to gauge the amount of Photofrin in the remaining tissue. The arterial concentration is modeled via a differential equation that describes how Photofrin accumulates and degrades in blood-plasma. We let CI be the plasma concentration during injection, k be the rate of elimination, and ko be the rate of infusion. The volume of distribution, V, for a particular drug is the volume of fluid that the drug would occupy if the total amount of drug in the body were at the same concentration as is present in the plasma. We model CI with the following first order, linear differential equation, dCI k0 = − kCI dt V 631 B. K. Singh, Paras Bhatnagar & Chitra Gupta Whose solution is CI = k0 1 − e − kt . Vk The plasma concentration after infusion is CA which exponentially decays. So assuming that the infusion ends at time τ , we have that dC A = −kC A , t ≥ τ , dt from which we conclude that C A = CI (τ ) e− kt = k0 1 − e− kτ e − k (t −τ ) , t ≥ τ . Vk The infusion rate ko is based on the fact that the drug is injected at 2mg/kg over a period of 5 minutes. Using an average 70kg adult, we have that 140mg of Photofrin are delivered in 5 minutes, and thus ko is 1680 mg/h. The mean volume of distribution for Photofrin is 0.491/kg, and in the case of 70 kg person, V is 34.3L. The rate of elimination, k is calculated by solving, t1 2 = In ( 2 ) k ; Where the half–life for Photofrin is t1 2 = 516 hours. k = 0.00134h –1. Using these Pharmacokinetic parameters for Photofrin and an infusion time of τ Hence = .083 h (5min), we have for t ≥ .083 that. The plasma concentration is CA = 4.08e – .00134 (t–.083) We convert CA to Molarity by dividing the concentrations by the molar mass of Photofrin (596798.4 mg/mol), which means we redefine CA to be CA = ( 1 −.00134( t −.083) 4.08e 596798.4 ) We use the arterial concentration CA to model Photofrin's concentration within a region that contains both arterial and non arterial tissues. We adapt the blood and tissue flow equations in [7], where a two-stage blood flow model sufficiently estimates tissue concentration of radioactively labeled water. We let Cp (t) be the Photofrin concentration at time t in the non-arterial tissue and Cporph(t) be the photofrin Concentration in the entire region. We let k1 and k2 be the tracer rate constants that describes the flow from tissue to blood and from blood to tissue; respectively. Allowing VA to be the percentage of arterial tissue in a region, we use the following model to calculate a region’s concentration at time T, T T C p (T ) = k1 ∫ C A ( t ) dt − k2 ∫ C p ( t ) dt 0 0 CPorph (T ) = VAC A (T ) + C p (T ) …… (1) We assume that the rates k1 and k2 are the same (k1 = k2 = 1). The time interval is divided into equal time steps of length Photofrin concentration, ∆t , and we have from [7] that Model (1) leads to the following approximate CPorph (T ) ≈ T T −∆t VAC A (T ) + ( k1 + VAk2 ) ∫ C A ( t ) dt − k2 ∫ CPorph ( t ) dt + ∆tCPorph (T − ∆t ) 2 0 0 1 + k2 ∆t / 2 This approximation is exact as OPTIMIZATION MODEL: – ∆t ↓ 0. 632 PHARMACOKINETIC AND OPTIMIZATION MODEL… In this part we design an optimization model that is based on the rates α ( p , a ,t ) to be the rate at which Photofrin is activated in region p by a light source focused along angle a at time t α ( p ,a ,t ) x( a ,t ) We point out that our problem is similar to that of optimally designing radiotherapy treatments, for which a substantial literature already exists [1, 4, 9]. From an optimization perspective, the problems are nearly identical, and we modify the model in [9] to meet our needs. However, radiotherapy uses highenergy particle physics to model how ionizing radiation damages cells, whereas PDT uses biochemical models to understand how an activated photosensitizer destroys tissue. If X(a, t) be the time (in seconds) that a light source is directed along angle a in time period t. Since α ( p , a ,t ) is the rate, at which the concentration of activated Photofrin accumulates in regions p, we have that the concentration of activated Photofrin accumulates in regions p is ∑( a ,t ) α ( p ,a ,t ) x( a ,t ) . Notice that we are assuming an additive accumulation of activated Photofrin, which means that we are ignoring cell regeneration during treatment. Cell growth and division have been investigated in the radiotherapy literature, where it is understood that cellular repair is negligible compared to the damage observed treatment. We from a dose matrix, A, from the rate and the columns by α ( p , a ,t ) by allowing the rows of A to be indexed by p ( a, t ) . Allowing x to be the vector of x (a,t) where the indices correspond to the columns of A, we have that x a Ax is the linear operator that takes exposure times, x, and maps them to concentrations of activated Photofrin, Ax. We partition the rows of in the following way, AT ← Tumor dose points A = AC ← Critical dose points AN ← Normal dose points With this notation we have that AT x, Ax and AN x are the concentrations of activated Photofrin in the tumorous, critical, and normal tissues, respectively. One of our goals is to decide exposure times that best treat the patient. We assume that the patient image is divided into a 64 × 64 grid (so a patient image has 4096 regions). CONCLUSION: – Our model a photosonsitizer’s blood and tissues concentration show that optimization can aid in the design and delivery of treatments, and investigate how increased localization in cancerous cells affects a treatment’s success. As one would expect, magnifying a photo sensitizer’s concentration in timorous tissue improves the optimal treatment. REFERENCES: – 1. Bartolozzi, F.; et al. Operational research techniques in medical treatment and diagnosis, a review. European Journal of Operations, research 121 (3): 435-466 2000. 2. Bourne, D.W.A., Triggs, E.J. and Eadie, M.J.; Pharmacokinetics for the nonmathematical. MTP Press Limited, 1986. 3. Chwilkowska, A. Scazko, J. Modrzycka, T. Marcinkowska, A., Malarska A., Bielwicz J., A., Patalas, J.D. and Banas, T.; Uptake of photofrin ii, a photo sensitizer used in photodynamic therapy, by tumor cells in vivo. Acta Biochemica polonica, 50(2): 509-513, 2003. 4. Holder, A.; Radiotherapy treatment design and linear programming. Technical Report 70, Trinity University mathematics, San Antonio, TX, 2002 to appear in the Handbook of Operations Research/ Management Science Applications in Health Care. 5. McMurry, J.; Organic Chemistry. Brocks/Cole 5th edition edition, 2000. 633 B. K. Singh, Paras Bhatnagar & Chitra Gupta 6. 7. 8. 9. Oikonen, V.; Model equations for [O] h2O Pet perfusion blood flow studies. Technical Report TPCMOD0004, Turka PET Centre Modeling, Turku University Central Hospital, 2002. Ravindra K.; Pandey and Gang Zheng Eorphyrins as photosensitizes in photodynamic therapy. In K.M. Kadish, K.M. smith, and R.Guilard, editors, The Porphyrin Handbook, volume 6, Academic Press 2000. Rosen, I., Lane, S., Morrill, S. and Belli, J.; Treatment plan optimization using linear programming. Medical Physics, 18(2): 141-152, 1991. Shepard, D. Ferris, M., Olivera, G. and Mackie, T.; Optimizing the delivery of radiation therapy to cancer patients. SIAM Review, 41 (4): 721-744, 1999. 634